1. ## Dice Probability

A fair n-sided die is rolled n times. Assuming the rolls are independent, calculate the probability of getting a match on roll i, i.e. on roll i the die shows i.

2. Originally Posted by gtaplayr
A fair n-sided die is rolled n times. Assuming the rolls are independent, calculate the probability of getting a match on roll i, i.e. on roll i the die shows i.
I don’t think you have posted what you are really asking.
Suppose we roll an ordinary die six times.
The probability of getting a 1 on the first roll is $\frac{1}{6}$.
The probability of getting a 2 on the second roll is $\frac{1}{6}$.

The probability of getting a 6 on the sixth roll is $\frac{1}{6}$.

Now is that truly what you asked?

3. Yes i understand that the likelihood of a matct is 1/n on any particular trial.
But i dont understand how to calulate the probability of getting a match on roll i.
Where does n start and where does it end?
If its infinite then isnt the probabilty of getting a match 1?

4. Originally Posted by gtaplayr
Yes i understand that the likelihood of a matct is 1/n on any particular trial.
But i dont understand how to calulate the probability of getting a match on roll i.
Where does n start and where does it end?
If its infinite then isnt the probabilty of getting a match 1?
Suppose we roll an ordinary die six times.
If you want the probability if getting the string 123456 then that is $\left(\frac{1}{6}\right)^6$.

For an n-sided die rolled n times, the probability if getting the string 123…n is $\left(\frac{1}{n}\right)^n$.

5. Originally Posted by gtaplayr
A fair n-sided die is rolled n times. Assuming the rolls are independent, calculate the probability of getting a match on roll i, i.e. on roll i the die shows i.
Perhaps you mean what is the probability of getting at least one match?

If so, this is 1 - P(no match). The probability of not getting a match on the nth roll is $1-1/n$, so assuming the rolls are independent, the probability of no macth in n rolls is $(1-1/n)^n$, and

$P(\text{at least one match}) = 1 - (1 - 1/n)^n$.

If you are interested in large n,

$\lim_{n \to \infty} 1 - (1 - 1/n)^n = 1 - 1/e$.